nLab IPC-property


The IPC-property (“inductive limit product commutation property”) is a technical condition on a category AA which ensures that presheaves with values in AA have a good notion of sheafification.


A category AA is said to satisfy the IPC-property if

  • it admits small products and filtered colimits;

  • for

    • every family {I s} sS\{I_s\}_{s \in S} of small filtered categories

    • and for any family {α s:I sA}\{\alpha_s : I_s \to A\} of functors

    • indexed by a small set SS

    the canonical morphism

    colim( sI sk sα s(π s(j))A) scolimα s colim \left( \prod_s I_s \stackrel{k \mapsto \prod_s \alpha_s(\pi_s(j))}{\to} A \right) \to \prod_s colim \alpha_s

    is an isomorphism.


The IPC property is definition 3.1.10 in

It is invoked for sheafification in section 17.4 there.

Last revised on December 31, 2023 at 19:25:58. See the history of this page for a list of all contributions to it.